The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. In 1927 it appeared in a second edition with an important Introduction To the Second Edition, an Appendix A that replaced *9 and an all-new Appendix C.
PM, as it is often abbreviated (not to be confused with Russell's 1903 Principles of Mathematics), is an attempt to derive all mathematical truths from a well-defined set of axioms and inference rules in symbolic logic. One of the main inspirations and motivations for PM was Frege's earlier work on logic, which had led to paradoxes discovered by Russell. These were avoided in PM by building an elaborate system of types: a set of elements is of a different type than is each of its elements (set is not the element; one element is not the set) and one cannot speak of the "set of all sets" and similar constructs, which lead to paradoxes (see Russell's paradox).
PM is widely considered by specialists in the subject to be one of the most important and seminal works in mathematical logic and philosophy since Aristotle's Organon. The Modern Library placed it 23rd in a list of the top 100 English-language nonfiction books of the twentieth century.
Scope of foundations laid
The Principia covered only set theory, cardinal numbers, ordinal numbers, and real numbers. Deeper theorems from real analysis were not included, but by the end of the third volume it was clear to experts that a large amount of known mathematics could in principle be developed in the adopted formalism. It was also clear how lengthy such a development would be.
A fourth volume on the foundations of geometry had been planned, but the authors admitted to intellectual exhaustion upon completion of the third.
Consistency and criticisms
According to Carnap's "Logicist Foundations of Mathematics", Russell wanted a theory that could plausibly be said to derive all of mathematics from purely logical axioms. However, Principia Mathematica required, in addition to the basic axioms of type theory, three further axioms that seemed to not be true as mere matters of logic, namely the axiom of infinity, the axiom of choice, and the axiom of reducibility. Since the first two were existential axioms, Russell phrased mathematical statements depending on them as conditionals. But reducibility was required to be sure that the formal statements even properly express statements of real analysis, so that statements depending on it could not be reformulated as conditionals. Frank P. Ramsey tried to argue that Russell's ramification of the theory of types was unnecessary, so that reducibility could be removed, but these arguments seemed inconclusive.
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